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A Technique for the Numerical Solution of Certain Integral Equations of the First Kind

1962-01-01
David Lee Phillips
article Free Access Share on A Technique for the Numerical Solution of Certain Integral Equations of the First Kind Author: David L. Phillips Argonne National Laboratory, Argonne, Illinois Argonne National … article Free Access Share on A Technique for the Numerical Solution of Certain Integral Equations of the First Kind Author: David L. Phillips Argonne National Laboratory, Argonne, Illinois Argonne National Laboratory, Argonne, IllinoisView Profile Authors Info & Claims Journal of the ACMVolume 9Issue 1pp 84–97https://doi.org/10.1145/321105.321114Published:01 January 1962Publication History 1,551citation3,679DownloadsMetricsTotal Citations1,551Total Downloads3,679Last 12 Months377Last 6 weeks38 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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Defective Nanographenes Containing Seven-Five-Seven (7–5–7)-Membered Rings

2021-01-27
Yiyang Fei , Yubin Fu , Xueqin Bai +7 more
Defects have been observed in graphene and are expected to play a key role in its optical, electronic, and magnetic properties. However, because most of the studies focused on the … Defects have been observed in graphene and are expected to play a key role in its optical, electronic, and magnetic properties. However, because most of the studies focused on the structural characterization, the implications of topological defects on the physicochemical properties of graphene remain poorly understood. Here, we demonstrate a bottom-up synthesis of three novel nanographenes (1–3) with well-defined defects in which seven-five-seven (7–5–7)-membered rings were introduced to their sp2 carbon frameworks. From the X-ray crystallographic analysis, compound 1 adopts a nearly planar structure. Compound 2, with an additional five-membered ring compared to 1, possesses a slightly saddle-shaped geometry. Compound 3, which can be regarded as the "head-to-head" fusion of 1 with two bonds, features two saddles connected together. The resultant defective nanographenes 1–3 were well-investigated by UV–vis absorption, cyclic voltammetry, and time-resolved absorption spectra and further corroborated by density functional theory (DFT) calculations. Detailed experimental and theoretical investigations elucidate that these three nanographenes 1–3 exhibit an anti-aromatic character in their ground states and display a high stability under ambient conditions, which contrast with the reported unstable biradicaloid nanographenes that contain heptagons. Our work reported herein offers insights into the understanding of structure-related properties and enables the control of the electronic structures of expanded nanographenes with atomically precise defects.

Are Adenine Strands Helical H-Aggregates?

2007-09-20
Lihong Hu , Yang Zhao , Fan Wang +4 more
On the basis of our quantum mechanical calculation, we propose that homogeneous single-stranded adenine bases (Ade-DNA) form helical H aggregates, and the photoexcited states can be described as Frenkel excitons. … On the basis of our quantum mechanical calculation, we propose that homogeneous single-stranded adenine bases (Ade-DNA) form helical H aggregates, and the photoexcited states can be described as Frenkel excitons. The calculated excitonic coupling between adjacent transition dipoles is in good agreement with the measured absorption spectrum of 20-base homogeneous adenine stacks that exhibits a blue shift of 2.6 nm relative to that of the monomeric species.

The logarithmic error and Newton's method for the square root

1969-02-01
Richard F. King , David Lee Phillips
The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is … The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is used as the measure of goodness of fit, optimal results are not obtained when the inital approximation is a best fit. It is shown here that if, instead, the so-called logarithmic error is used, then a best initial fit is optimal for both types of error. Moreover, use of the logarithmic error appears to simplify the problem of determining the optimal initial approximation.
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A note on best one-sided approximations

1971-09-01
David Lee Phillips
In this note we consider the relationship between best approximations and best one-sided approximations for three different measures of goodness of fit. For these measures simple relationships exist between best … In this note we consider the relationship between best approximations and best one-sided approximations for three different measures of goodness of fit. For these measures simple relationships exist between best approximations and best one-sided approximations. In particular it is shown that a best approximation and best one-sided approximation differ only by a multiplicative constant when the measure is the uniform norm of the relative error. In this case problems involving best one-sided approximations can be reduced to problems involving best approximations. The result is especially significant if one wants to numerically determine a best one-sided approximation, since algorithms exist for numerically determining best approximations when the measure is the uniform norm of the relative error (see, for example, [1]).
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AREA RELATIONS BETWEEN LABELING INDEX CURVES FROM MULTICOMPARTMENT CELL KINETICS MODELS.

1970-01-01
R. N. Buchal , David Lee Phillips , Ernesto Trucco
ABSTRACT The study of the multicompartment models in cell kinetics can be simplified by the use of a conservation law relating the integrals, with respect to time, of labeling index … ABSTRACT The study of the multicompartment models in cell kinetics can be simplified by the use of a conservation law relating the integrals, with respect to time, of labeling index in various compartments. We present a proof of the conservation law directed to the nonmathematician, as well as four applications from biology. The first demonstrates a contradiction, which biologists have yet to resolve, between a certain biological model and experimental results. The second and third are simpler proofs of results already proven by other techniques in the literature. The fourth is a result which appears to be new.

Are Adenine Strands Helical H-aggregates?

2006-01-01
Yang Zhao , Lihong Hu , Fan Wang +4 more
This paper has been withdrawn by the authors due to request from a JPCB editor. This paper has been withdrawn by the authors due to request from a JPCB editor.

Mathematical Notes

1968-05-01
Henryk Minc , Richard C. Moore , E. J. Vought +7 more

Rational Design and Reaction Mechanism Study of the Photochemical Rearrangement of Fulvene Derivatives

2025-05-13
Shuang Xu , Tianhe Yang , Shulin Zhang +6 more
Photochemistry is considered one of the most efficient and reproducible techniques in organic synthesis. Recently, List and co-workers reported an efficient UV light triggered photochemical synthesis of spiro[2,4]heptadiene from fulvenes … Photochemistry is considered one of the most efficient and reproducible techniques in organic synthesis. Recently, List and co-workers reported an efficient UV light triggered photochemical synthesis of spiro[2,4]heptadiene from fulvenes with different substituents ( Angew. Chem., Int. Ed. 2023, 62, e202303119); however, the mechanistic details remain unclear, and the intermediates have not been characterized. To facilitate the applications of this novel photochemical reaction, we theoretically designed a series of fulvene derivatives with different parent molecular skeletons for analyzing the substitution effects, and two of the representative fulvenes were synthesized for investigating the reaction mechanisms by employing time-resolved transient absorption spectroscopy (TA) experiments. It has been found that instead of density functional theory, the second-order n-electron valence state perturbation theory is necessary to acquire reliable theoretical characterization of the fulvenes examined. Our designed fulvenes were found to undergo the photorearrangement cyclopropanation reaction on the basis of photoproduct analysis. The intermediate species involved in the intramolecular hydrogen atom transfer and cyclization processes within the photorearrangement reaction were characterized by TA spectroscopy, and the full reaction pathways were proposed. Our work not only reveals the detailed mechanism of this photorearrangement reaction but also demonstrates the significance of appropriate theoretical methods for rational molecular design.

Rational Design and Reaction Mechanism Study of the Photochemical Rearrangement of Fulvene Derivatives

2025-05-13
Shuang Xu , Tianhe Yang , Shulin Zhang +6 more
Photochemistry is considered one of the most efficient and reproducible techniques in organic synthesis. Recently, List and co-workers reported an efficient UV light triggered photochemical synthesis of spiro[2,4]heptadiene from fulvenes … Photochemistry is considered one of the most efficient and reproducible techniques in organic synthesis. Recently, List and co-workers reported an efficient UV light triggered photochemical synthesis of spiro[2,4]heptadiene from fulvenes with different substituents ( Angew. Chem., Int. Ed. 2023, 62, e202303119); however, the mechanistic details remain unclear, and the intermediates have not been characterized. To facilitate the applications of this novel photochemical reaction, we theoretically designed a series of fulvene derivatives with different parent molecular skeletons for analyzing the substitution effects, and two of the representative fulvenes were synthesized for investigating the reaction mechanisms by employing time-resolved transient absorption spectroscopy (TA) experiments. It has been found that instead of density functional theory, the second-order n-electron valence state perturbation theory is necessary to acquire reliable theoretical characterization of the fulvenes examined. Our designed fulvenes were found to undergo the photorearrangement cyclopropanation reaction on the basis of photoproduct analysis. The intermediate species involved in the intramolecular hydrogen atom transfer and cyclization processes within the photorearrangement reaction were characterized by TA spectroscopy, and the full reaction pathways were proposed. Our work not only reveals the detailed mechanism of this photorearrangement reaction but also demonstrates the significance of appropriate theoretical methods for rational molecular design.

Defective Nanographenes Containing Seven-Five-Seven (7–5–7)-Membered Rings

2021-01-27
Yiyang Fei , Yubin Fu , Xueqin Bai +7 more
Defects have been observed in graphene and are expected to play a key role in its optical, electronic, and magnetic properties. However, because most of the studies focused on the … Defects have been observed in graphene and are expected to play a key role in its optical, electronic, and magnetic properties. However, because most of the studies focused on the structural characterization, the implications of topological defects on the physicochemical properties of graphene remain poorly understood. Here, we demonstrate a bottom-up synthesis of three novel nanographenes (1–3) with well-defined defects in which seven-five-seven (7–5–7)-membered rings were introduced to their sp2 carbon frameworks. From the X-ray crystallographic analysis, compound 1 adopts a nearly planar structure. Compound 2, with an additional five-membered ring compared to 1, possesses a slightly saddle-shaped geometry. Compound 3, which can be regarded as the "head-to-head" fusion of 1 with two bonds, features two saddles connected together. The resultant defective nanographenes 1–3 were well-investigated by UV–vis absorption, cyclic voltammetry, and time-resolved absorption spectra and further corroborated by density functional theory (DFT) calculations. Detailed experimental and theoretical investigations elucidate that these three nanographenes 1–3 exhibit an anti-aromatic character in their ground states and display a high stability under ambient conditions, which contrast with the reported unstable biradicaloid nanographenes that contain heptagons. Our work reported herein offers insights into the understanding of structure-related properties and enables the control of the electronic structures of expanded nanographenes with atomically precise defects.

Are Adenine Strands Helical H-Aggregates?

2007-09-20
Lihong Hu , Yang Zhao , Fan Wang +4 more
On the basis of our quantum mechanical calculation, we propose that homogeneous single-stranded adenine bases (Ade-DNA) form helical H aggregates, and the photoexcited states can be described as Frenkel excitons. … On the basis of our quantum mechanical calculation, we propose that homogeneous single-stranded adenine bases (Ade-DNA) form helical H aggregates, and the photoexcited states can be described as Frenkel excitons. The calculated excitonic coupling between adjacent transition dipoles is in good agreement with the measured absorption spectrum of 20-base homogeneous adenine stacks that exhibits a blue shift of 2.6 nm relative to that of the monomeric species.

Are Adenine Strands Helical H-aggregates?

2006-01-01
Yang Zhao , Lihong Hu , Fan Wang +4 more
This paper has been withdrawn by the authors due to request from a JPCB editor. This paper has been withdrawn by the authors due to request from a JPCB editor.

A note on best one-sided approximations

1971-09-01
David Lee Phillips
In this note we consider the relationship between best approximations and best one-sided approximations for three different measures of goodness of fit. For these measures simple relationships exist between best … In this note we consider the relationship between best approximations and best one-sided approximations for three different measures of goodness of fit. For these measures simple relationships exist between best approximations and best one-sided approximations. In particular it is shown that a best approximation and best one-sided approximation differ only by a multiplicative constant when the measure is the uniform norm of the relative error. In this case problems involving best one-sided approximations can be reduced to problems involving best approximations. The result is especially significant if one wants to numerically determine a best one-sided approximation, since algorithms exist for numerically determining best approximations when the measure is the uniform norm of the relative error (see, for example, [1]).
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AREA RELATIONS BETWEEN LABELING INDEX CURVES FROM MULTICOMPARTMENT CELL KINETICS MODELS.

1970-01-01
R. N. Buchal , David Lee Phillips , Ernesto Trucco
ABSTRACT The study of the multicompartment models in cell kinetics can be simplified by the use of a conservation law relating the integrals, with respect to time, of labeling index … ABSTRACT The study of the multicompartment models in cell kinetics can be simplified by the use of a conservation law relating the integrals, with respect to time, of labeling index in various compartments. We present a proof of the conservation law directed to the nonmathematician, as well as four applications from biology. The first demonstrates a contradiction, which biologists have yet to resolve, between a certain biological model and experimental results. The second and third are simpler proofs of results already proven by other techniques in the literature. The fourth is a result which appears to be new.

The logarithmic error and Newton's method for the square root

1969-02-01
Richard F. King , David Lee Phillips
The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is … The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is used as the measure of goodness of fit, optimal results are not obtained when the inital approximation is a best fit. It is shown here that if, instead, the so-called logarithmic error is used, then a best initial fit is optimal for both types of error. Moreover, use of the logarithmic error appears to simplify the problem of determining the optimal initial approximation.
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Mathematical Notes

1968-05-01
Henryk Minc , Richard C. Moore , E. J. Vought +7 more

A Technique for the Numerical Solution of Certain Integral Equations of the First Kind

1962-01-01
David Lee Phillips
article Free Access Share on A Technique for the Numerical Solution of Certain Integral Equations of the First Kind Author: David L. Phillips Argonne National Laboratory, Argonne, Illinois Argonne National … article Free Access Share on A Technique for the Numerical Solution of Certain Integral Equations of the First Kind Author: David L. Phillips Argonne National Laboratory, Argonne, Illinois Argonne National Laboratory, Argonne, IllinoisView Profile Authors Info & Claims Journal of the ACMVolume 9Issue 1pp 84–97https://doi.org/10.1145/321105.321114Published:01 January 1962Publication History 1,551citation3,679DownloadsMetricsTotal Citations1,551Total Downloads3,679Last 12 Months377Last 6 weeks38 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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Coauthor Papers Together
Wai‐Ming Kwok 2
Chensheng Ma 2
Lihong Hu 2
Robert J. Nielsen 1
Hudson V. Kronk 1
Yiyang Fei 1
Yubin Fu 1
Ernesto Trucco 1
Richard F. King 1
Al Somayajulu 1
R. N. Buchal 1
Yu Fang 1
Shuang Xu 1
Hartmut Komber 1
Junzhi Liu 1
Tianhe Yang 1
Shengqiang Zhou 1
Z. Z. Yeh 1
Richard C. Moore 1
B. J. Pettis 1
E. J. Vought 1
Rachel Cox 1
Jiani Ma 1
Dale E. Varberg 1
GuanHua Chen 1
Xueqin Bai 1
Lili Du 1
C. Sloyer 1
R. E. Smithson 1
N.M. Rice 1
Klaus Eldridge 1
Fan Wang 1
Fan Wang 1
Yang Zhao 1
Shulin Zhang 1
Xinliang Feng 1
GuanHua Chen 1
Zichao Li 1
Yifan Su 1
Le Yu 1
Yang Zhao 1
Henryk Minc 1
Kam‐Hung Low 1
William Feller 1
Tim Robertson 1
Yang Cheng 1

Computational Aspects of Chebyshev Approximation Using a Generalized Weight Function

1968-03-01
David Moursund
Previous article Next article Computational Aspects of Chebyshev Approximation Using a Generalized Weight FunctionD. G. MoursundD. G. Moursundhttps://doi.org/10.1137/0705010PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] David G. Moursund, Chebyshev approximation using … Previous article Next article Computational Aspects of Chebyshev Approximation Using a Generalized Weight FunctionD. G. MoursundD. G. Moursundhttps://doi.org/10.1137/0705010PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] David G. Moursund, Chebyshev approximation using a generalized weight function, SIAM J. Numer. Anal., 3 (1966), 435–450 10.1137/0703038 MR0204936 0163.07302 LinkGoogle Scholar[2] David G. Moursund, Optimal starting values for Newton-Raphson calculation of $\surd x$, Comm. ACM, 10 (1967), 430–432 10.1145/363427.363454 MR0240952 CrossrefISIGoogle Scholar[3] D. G. Moursund and , G. D. Taylor, Optimal starting values for the Newton-Raphson calculation of inverses of certain functions, SIAM J. Numer. Anal., 5 (1968), 138–150 10.1137/0705011 MR0225481 0164.17401 LinkISIGoogle Scholar[4] John R. Rice, The approximation of functions. Vol. I: Linear theory, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964xi+203 MR0166520 0114.27001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails The best interpolating approximation and weighted approximationJournal of Approximation Theory, Vol. 35, No. 1 Cross Ref Vergleich zwischen Diskretisierungsverfahren und parametrischen Methoden an einfachen TestbeispielenZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 56, No. 1 Cross Ref Simultanapproximation bei Randwertaufgaben Cross Ref A Linear Remes-Type Algorithm for Relative Error ApproximationMartha Ann Griesel14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 11, No. 1AbstractPDF (395 KB)Computing fourier integrals by means of near-optimal quadrature rules of the Lagrangian typeComputing, Vol. 11, No. 1 Cross Ref An Improved Newton Iteration for Calculating Roots which is Optimal Cross Ref Optimal starting approximations for generating square root for slow or no divideCommunications of the ACM, Vol. 13, No. 9 Cross Ref A Survey of Practical Rational and Polynomial Approximation of FunctionsW. J. Cody18 July 2006 | SIAM Review, Vol. 12, No. 3AbstractPDF (2298 KB)Some Remarks on a Paper of D. G. MoursundL. Wuytack14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 7, No. 2AbstractPDF (426 KB)On an algorithm for nonlinear minimax approximationCommunications of the ACM, Vol. 13, No. 3 Cross Ref Chebyshev approximation with respect to a weight functionJournal of Approximation Theory, Vol. 2, No. 3 Cross Ref On Approximations by Polynomials Having Restricted Ranges. IIL. L. Schumaker and G. D. Taylor14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 6, No. 1AbstractPDF (536 KB)Uniform Rational Approximation Using a Generalized Weight FunctionD. G. Moursund and G. D. Taylor14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 4AbstractPDF (647 KB)Uniform rational weighted approximations having restricted rangesJournal of Approximation Theory, Vol. 1, No. 4 Cross Ref On Approximation by Polynomials Having Restricted RangesG. D. Taylor3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 2AbstractPDF (1067 KB)Optimal Starting Values for the Newton-Raphson Calculation of Inverses of Certain FunctionsD. G. Moursund and G. D. Taylor14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 1AbstractPDF (1155 KB) Volume 5, Issue 1| 1968SIAM Journal on Numerical Analysis History Submitted:13 March 1967Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705010Article page range:pp. 126-137ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Discrete Analogues of the Dirichlet Problem with Isolated Singularities

1968-03-01
James H. Bramble , B. E. Hubbard , Milos̋ Zlámal
* Received by the editors December 7, 1966. t Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland 20742. The research of the first author was … * Received by the editors December 7, 1966. t Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland 20742. The research of the first author was supported in part by the National Science Foundation under Grant NSF GP-3666, that of the second author supported in part by the Atomic Energy Commission under Grant AEC-AT(40-1) 3443, and that of the third supported in part by the National Science Foundation under Grant NSF GP-4291.

The logarithmic error and Newton's method for the square root

1969-02-01
Richard F. King , David Lee Phillips
The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is … The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is used as the measure of goodness of fit, optimal results are not obtained when the inital approximation is a best fit. It is shown here that if, instead, the so-called logarithmic error is used, then a best initial fit is optimal for both types of error. Moreover, use of the logarithmic error appears to simplify the problem of determining the optimal initial approximation.
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Rings of Continuous Functions

1960-01-01
Leonard Gillman , Meyer Jerison

On Estimating a Density which is Measurable with Respect to a $\sigma$-Lattice

1967-04-01
Tim Robertson
This paper is concerned with the problem of estimating a probability density which is known to be measurable with respect to a $\sigma$-lattice of subsets of the space on which … This paper is concerned with the problem of estimating a probability density which is known to be measurable with respect to a $\sigma$-lattice of subsets of the space on which it is defined. Our solution is represented as a conditional expectation. (Generally in the literature "conditional expectation" refers to "conditional expectation given a $\sigma$-field." However since we shall be concerned exclusively with "conditional expectation given a $\sigma$-lattice" we shall use this abbreviated terminology for the latter, more general concept.) Brunk [2] discusses conditional expectations and many of the extremum problems for which they provide solutions. We shall consider the case where the measure space, $(\Omega, \mathscr{A}, \mu)$ on which the density is defined is totally finite. Let $\mathscr{L}$ denote a $\sigma$-lattice of subsets of $\Omega(\mathscr{L} \subset \mathcal{A})$. A $\sigma$-lattice, by definition, is closed under countable unions and intersections and contains both $\Omega$ and the null set $\varnothing$. Let $\omega_1, \omega_2, \cdots, \omega_n$ be a sample of independent observations chosen in $\Omega$ according to the unknown, $\mathscr{L}$-measurable density $f$. We say that a point is chosen in $\Omega$ according to $f$ if the probability that it will lie in any set $A$ in $\mathscr{A}$ is given by $\int_Af d\mu$. The function $f$ is $\mathscr{L}$-measurable if the set $\lbrack f > a\rbrack$ is in $\mathscr{L}$ for each real number $a$. We shall use the maximum likelihood criterion for choosing an estimate. In other words we wish to find an $\mathscr{L}$-measurable density $\hat f$ such that the product of the values of $\hat f$ at the observed points is at least as large as the product of the values of any other $\mathscr{L}$-measurable density at those points. Such a function will be called a maximizing function. Clearly the $\sigma$-lattice $\mathscr{L}$ must satisfy some restrictions in order for the problem to be of any interest at all. For example if $(\Omega, \mathscr{A}, \mu)$ is a finite subinterval of the real line together with Borel subsets and Lebesgue measure and if $\mathscr{L} = \mathscr{A}$ then there are many obvious solutions if the density is bounded, and none at all if it is not bounded. The second section of this paper is devoted to the restrictions that we impose and to showing that these restrictions are satisfied in some problems which are of interest. The fourth section of this paper is devoted to some results on the asymptotic properties of our estimates in three special cases. The methods used are similar to those used by Marshall and Proschan [4]. The final section contains some observations on the problem of estimating a density on a non-totally finite measure space. Let $L_2$ denote the set of square integrable random variables and $L_2(\mathscr{L})$ the collection of all those members of $L_2$ which are $\mathscr{L}$-measurable. Let $R(\mathscr{L})$ denote the collection of all $\mathscr{L}$-measurable random variables. Let $\mathscr{B}$ denote the collection of Borel subsets of the real line. We shall adopt the following definition for the conditional expectation, $E_\mu(f \mid \mathscr{L})$, of a random variable given a $\sigma$-lattice. DEFINITION 1.1. If $f \epsilon L_2$ then $g \epsilon L_2(\mathscr{L})$ is equal to $E_\mu(f \mid \mathscr{L})$ if and only if $g$ has both of the following properties: \begin{equation*}\tag{1.1}\int (f - g)h d\mu \leqq 0 \text{for all} h \epsilon L_2(\mathscr{L})\end{equation*} and \begin{equation*}\tag{1.2}\int_B(f - g)d\mu = 0\quad\text{for all}\quad B \epsilon g^{-1}(\mathscr{B})\end{equation*}. (Brunk [1] shows that there is such a random variable $g$ associated with each $f \epsilon L_2$ and that $g$ is unique in the sense that if $g'$ is any other member of $L_2(\mathscr{L})$ having these properties then $g = g'\lbrack\mu\rbrack$.) In order to motivate the consideration of a problem such as the one we take up in this paper we introduce the following examples. The first example is discussed in [5]. EXAMPLE 1.1. Suppose $\Omega$ is a finite set and we denote its elements by $1, 2, \cdots, k$. Let $\mathscr{A}$ be the collection of all subsets of $\Omega$ and suppose $\mu$ assigns positive mass to each point in $\Omega$. Suppose $\mathscr{L}$ is an arbitrary $\sigma$-lattice of subsets of $\Omega$. Let $n_i$ denote the number of times the point $i$ is observed. We wish to find an $\mathscr{L}$-measurable density $\hat f$ on $\Omega$ such that $\prod^k_{i = 1} \hat f(i)^{n_i} \geqq \prod^k_{i = 1} h(i)^{n_i}$ for every other $\mathscr{L}$-measurable density $h$. It is shown in [5] that a solution is given by $\hat f = E_\mu(g \mid \mathscr{L})$ where $g(i) = n_i\lbrack n \cdot \mu(i)\rbrack^{-1}$. The problem posed in the next example was solved by Pyke (personal communication with Professor Brunk) and for a monotone density by Grenander [3]. EXAMPLE 1.2. Suppose $\Omega$ is a closed subinterval of the real line $(\Omega = \lbrack c, d\rbrack), \mathscr{a}$ is the collection of Borel subsets of $\Omega$ and $\mu$ is Lebesque measure. We wish to estimate the density $f$ which is known to be unimodal at some unknown point in $\Omega$. Suppose that our observations are ordered: $\omega_1 < \omega_2 < \cdots < \omega_n$. If $h$ is any unimodal density with mode at $a$ and $\omega_j < a < \omega_{j + 1}$ then define the function $g$ by: $g(x) = h(\omega_j)\quad (\omega_j \leqq x < a) \\ = h(\omega_{j + 1}) \quad (a \leqq x \leqq \omega_{j + 1}) \\ = h(x) \quad \text{otherwise}$. It is easily seen that the density $\hat f = \lbrack \int g d\mu\rbrack^{-1} \cdot g$ is unimodal with mode at $\omega_j$ or $\omega_{j+1}$ and that the product of the values of $\hat f$ at the observed points is at least as large as the product of the corresponding values of $h$. Hence our problem reduces to finding an estimate which has mode at one of the observed points. Similarly we can show that any maximizing estimate must be constant on every open interval joining two consecutive observed values. The next remark is easy to verify. REMARK 1.1. Let $\mathscr{L}$ be the $\sigma$-lattice of subsets of $\Omega$ consisting of all those intervals containing the point $a$. A function $f$ on $\Omega$ is unimodal at $a$ if and only if $f$ is $\mathscr{L}$-measurable. If we can estimate the density subject to the restriction that it is unimodal at a fixed point then we can select an estimate by comparing the ones we get by assuming the mode is at particular observations. We see thus that the problem of estimating a unimodal density reduces to estimating a density which is measurable with respect to a $\sigma$-lattice and constant on intervals joining consecutive observations. We shall see that these results for a unimodal density are typical of a larger class of problems. Now consider the problem of estimating a unimodal density on the reals together with Lebesgue measure. Clearly any estimate must be zero outside of the smallest closed interval containing all the observed points. Hence this problem reduces to the one discussed above.
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The multiplicative groups of a ring

1965-01-01
H. K. Farahat

Generalized logarithmic error and Newton’s method for the $m$th root.

1970-01-01
David L. Phillips
The problem of obtaining optimal starting values for the calculation of integer roots using Newton’s method is considered. It has been shown elsewhere that if relative error is used as … The problem of obtaining optimal starting values for the calculation of integer roots using Newton’s method is considered. It has been shown elsewhere that if relative error is used as the measure of goodness of fit. then optimal results are not obtained when the initial approximation is a best fit. Furthermore, if the so-called logarithmic error instead of the relative error is used in the square root case, then a best initial fit is optimal for both errors It is shown here that for each positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-over-equals 3"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≧<!-- ≧ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m \geqq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and each negative integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a certain generalized logarithmic error for which a best initial fit to the mth root is optimal. It is then shown that an optimal fit can be found by just multiplying a best relative error fit by a certain constant. Also, explicit formulas are found for the optimal initial linear fit.
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Optimal Starting Approximations for Newton's Method

1969-04-01
P. H. Sterbenz , C. T. Fike
Various writers have dealt with the subject of optimal starting approxi- mations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead … Various writers have dealt with the subject of optimal starting approxi- mations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some pre- scribed form so that the maximum relative error after any number of Newton iterations is as small as possible. U 1. Introduction. The choice of polynomial and rational starting approximations for square-root calculation by Newton's method has been the subject of various investigations (e.g., (1)-(5)). These approach the problem from several different points of view. In this paper we will show how approximations obtained from these different viewpoints are related and how some of them can be derived from others. The problem of evaluating V/x for any x > 0 is easily reduced to the problem of evaluating V/x for x in some closed interval (a, b) such that 0 < a < b. Here (a, b) depends on the radix of the floating-point number system of the computer to be used; typical possibilities are (1/16, 1) and (-, 2). The following procedure is used to compute an approximate value for VIx. Using a polynomial or rational approxima- tion f(x) to V/x, valid in (a, b), compute a starting value yo = f(x) and then obtain Y1, Y22 .2 Yn by means of the relation Yk+1 = 12 (Yk + Xl8k), k = O. 1, * ,n - 1 . Then Yn V/x. It is customary not to test for convergence, since the number of iterations required in practice is quite small. Instead, the number of iterations n is

Continuously invertible spaces

1962-06-01
P. Doyle , J. G. Hocking
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A characterization of hereditarily indecomposable continua

1966-01-01
William R. Zame
A continuum (compact, connected Hausdorff space) is said to be indecomposable if it is not the sum of two proper subcontinua. It is said to be hereditarily indecomposable if each … A continuum (compact, connected Hausdorff space) is said to be indecomposable if it is not the sum of two proper subcontinua. It is said to be hereditarily indecomposable if each of its subcontinua is indecomposable. Knaster [3 ] showed the existence of such continua in the plane, and Bing [1], [2] showed that there exist hereditarily indecomposable continua of every positive dimension. In the present note we obtain a characterization of such continua by means of certain separation properties. If M is a continuum and p is a point of M, we recall that the composant of p in M, written Kp(M), is the union of all proper subcontinua of M containing p. Each composant of M is dense in M, and if M is indecomposable then its composants are pairwise disjoint [4]. Further, if the continuum X is not the sum of three continua, no one of which is contained in the sum of the other two, then X is either indecomposable or the sum of two indecomposable continua.
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Optimal Starting Values for the Newton-Raphson Calculation of Inverses of Certain Functions

1968-03-01
David Moursund , G. D. Taylor
Previous article Next article Optimal Starting Values for the Newton-Raphson Calculation of Inverses of Certain FunctionsD. G. Moursund and G. D. TaylorD. G. Moursund and G. D. Taylorhttps://doi.org/10.1137/0705011PDFBibTexSections ToolsAdd to … Previous article Next article Optimal Starting Values for the Newton-Raphson Calculation of Inverses of Certain FunctionsD. G. Moursund and G. D. TaylorD. G. Moursund and G. D. Taylorhttps://doi.org/10.1137/0705011PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] David G. Moursund, Chebyshev approximation using a generalized weight function, SIAM J. Numer. Anal., 3 (1966), 435–450 10.1137/0703038 MR0204936 0163.07302 LinkGoogle Scholar[2] David G. Moursund, Optimal starting values for Newton-Raphson calculation of $\surd x$, Comm. ACM, 10 (1967), 430–432 10.1145/363427.363454 MR0240952 CrossrefISIGoogle Scholar[3] D. G. Moursund, Computational aspects of Chebyshev approximation using a generalized weight function, SIAM J. Numer. Anal., 5 (1968), 126–137 10.1137/0705010 MR0229366 0164.18501 LinkISIGoogle Scholar[4] John R. Rice, The approximation of functions. Vol. I: Linear theory, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964xi+203 MR0166520 0114.27001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Applications of Transformation Theory: A Legacy from Zolotarev (1847–1878) Cross Ref Optimal partitioning of Newton’s method for calculating roots1 January 1980 | Mathematics of Computation, Vol. 35, No. 152 Cross Ref Numerische Methoden zur Behandlung einiger Problemklassen der nichtlinearen Tschebyscheff-Approximation mit NebenbedingungenNumerische Mathematik, Vol. 28, No. 1 Cross Ref Optimal rational starting approximationsJournal of Approximation Theory, Vol. 12, No. 2 Cross Ref Optimal starting approximations for iterative schemesJournal of Approximation Theory, Vol. 9, No. 1 Cross Ref An Improved Newton Iteration for Calculating Roots which is Optimal Cross Ref Kolmogoroff's criterion for constrained rational approximationJournal of Approximation Theory, Vol. 4, No. 2 Cross Ref Improved Newton iteration for integral roots1 January 1971 | Mathematics of Computation, Vol. 25, No. 114 Cross Ref Some Remarks on a Paper of D. G. MoursundL. Wuytack14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 7, No. 2AbstractPDF (426 KB)Optimal starting approximations for Newton's methodJournal of Approximation Theory, Vol. 3, No. 2 Cross Ref Chapter 5 The Bubnov-Galerkin Method Cross Ref Generalized logarithmic error and Newton’s method for the 𝑚th root.1 January 1970 | Mathematics of Computation, Vol. 24, No. 110 Cross Ref Uniform approximation of vector-valued functionsNumerische Mathematik, Vol. 13, No. 3 Cross Ref Nonlinear Chebychev approximation by vector-normsJournal of Approximation Theory, Vol. 2, No. 1 Cross Ref Uniform Rational Approximation Using a Generalized Weight FunctionD. G. Moursund and G. D. Taylor14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 4AbstractPDF (647 KB)Continuity of the best approximation operator for restricted range approximationsJournal of Approximation Theory, Vol. 1, No. 4 Cross Ref Uniform rational weighted approximations having restricted rangesJournal of Approximation Theory, Vol. 1, No. 4 Cross Ref On Approximation by Polynomials Having Restricted RangesG. D. Taylor3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 2AbstractPDF (1067 KB)Computational Aspects of Chebyshev Approximation Using a Generalized Weight FunctionD. G. Moursund14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 1AbstractPDF (953 KB) Volume 5, Issue 1| 1968SIAM Journal on Numerical Analysis History Submitted:01 May 1967Accepted:16 September 1967Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705011Article page range:pp. 138-150ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

On the decomposition by generations of the PLM-function

1970-01-01
Peter J. Brockwell , Ernesto Trucco

On an extension of the concept conditional expectation

1963-04-01
H. D. Brunk
The purpose of the present note is in part to extend results of Sidák [3] to cases where the measure is not finite, perhaps not even <r-finite.In particular, the conditional … The purpose of the present note is in part to extend results of Sidák [3] to cases where the measure is not finite, perhaps not even <r-finite.In particular, the conditional expectation of a random variable in Lx is obtained without the use of the Radon-Nikodym Theorem, which, indeed, does not apply with the required generality.It was found possible to extend at the same time results of [l] so as to omit the requirement that the measure be totally finite.For this reason the c-fields discussed in [3] are here replaced by cr-lattices.In this connection it may be observed that E(X\ £) is the solution of the regression problem: given X<ElL2, choose Y in the class C of random variables in L2 measurable with respect to a cr-field £ so as to minimize E(X-Y)2; E(X\ £) is the projection in L2 of X on C. Certain problems of maximum likelihood estimation of ordered parameters have solutions which also solve the above regression problem, in which £ is not a cr-field but is a cr-lattice, closed under countable union and countable intersection, but not necessarily under complementation (see references in [l]).Let (fi, S, jit) be a measure space: S is a c-field of subsets of fi, and ß is a measure, cr-additive and complete, but not necessarily cr-finite.
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Improved Newton iteration for integral roots

1971-01-01
Richard F. King
An improved Newton iteration procedure for computing <italic>p</italic>th roots from best Chebyshev or Moursund initial approximations is developed. It differs from the usual Newton method by a multiplicative factor at … An improved Newton iteration procedure for computing <italic>p</italic>th roots from best Chebyshev or Moursund initial approximations is developed. It differs from the usual Newton method by a multiplicative factor at each step. This multiplier halves the relative error by translating the usual one-sided error curve into a two-sided one, and then adjusting to make a Moursund-like fit. The generalized logarithmic error is used in determining this set of factors.
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Some remarks on integral equations with kernels: <i>L</i> (ξ <sub>1</sub> - <i>x</i> <sub>1</sub> ,..., ξ <sub>n</sub> - <i>x</i> <sub>n</sub> ; α)

1949-06-07
G. Kreisel
The paper discusses integral equations of the type k (ξ 1 ... ξ n ) = ∫ -∞ ∞ ... ∫ -∞ ∞ ψ ( x 1 ... x n … The paper discusses integral equations of the type k (ξ 1 ... ξ n ) = ∫ -∞ ∞ ... ∫ -∞ ∞ ψ ( x 1 ... x n ) L (ξ 1 -x 1 ... ξ n -x n ;α) d x 1 ... d x n , where L is L -integrable, and ψ and k are bounded. Since rapidly oscillating ψ have a small k , and since measurements of k are necessarily uncertain within non-zero limits of experimental error, very different ψ are consistent with any given set of measurements of k . Thus ψ is not determined by measurements of k . Instead of ψ, partial information about ψ that is not sensitive to rapid oscillations of ψ, can be obtained from k . In the present paper we consider smoothed versions of ψ, and their applications to gravity survey and the theory of surface waves. (1) For given L we construct normalized smoothing functions μ( x 1 ... x n ), so that ψ( x 1 ... x n ) = ∫ -∞ ∞ ... ∫ -∞ ∞ ψ( u 1 ... u n )μ (x 1 - u 1 ... x n - u n ) d u 1 .... d u n can be calculated from measurements of k( (ξ 1 ... ξ n ). The method is applied to gravity survey, where the distribution ψ of masses on a plane ∑' is to be calculated from the normal force k on another (parallel) plane∑. (2) By studying suitable smoothing functions we get a lower bound for the maximum modulus of the functions ψ which are consistent with given experimental values of k . The bound is large if k is known to vary rapidly. The bound is also large if α is large and if the Fourier transform of L tends to 0 when α→∞. The results are applied to gravity survey where now we consider the normal force on ∑ due to masses of bounded density distributed in the space below ∑', where ∑' is itself below ∑. If the normal force is not uniform the distance between ∑ and ∑' must not be too large, the estimate depending on the bound for the density. Also fairly general conditions are imposed on ψ so that ψ that is approximately determined by measurements of k , and an example from the theory of propagation in dispersive media is given where such conditions may be justified. The gist of the paper is contained in theorems A and B.
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Grains and grain boundaries in single-layer graphene atomic patchwork quilts

2011-01-01
Pinshane Y. Huang , Carlos Ruiz‐Vargas , Arend M. van der Zande +7 more
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Two-dimensional gas of massless Dirac fermions in graphene

2005-11-01
Kostya S. Novoselov , A. K. Geǐm , С. В. Морозов +5 more
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Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene

2009-10-14
Xu Du , Ivan Skachko , Fabian Duerr +2 more
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A Least Square Procedure for Solving Integral Equations by Polynomial Approximation

1941-04-01
F. B. Hildebrand , Prescott D. Crout

Axiomatic Treatment of Rank in Infinite Sets

1949-08-01
R. Rado
1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All … 1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered by Whitney are finite. In the present note the axiomatic treatment of rank is extended to sets of any cardinal. In the special case of algebraic dependence of elements of a field with respect to a sub-field, similar questions have already been considered by Steinitz [2].
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An Application of Polynomial Approximation to the Solution of Integral Equations Arising in Physical Problems

1940-04-01
Prescott D. Crout

Wave-shaped polycyclic hydrocarbons with controlled aromaticity

2019-01-01
Ji Ma , Ke Zhang , Karl Sebastian Schellhammer +7 more
Controlling the aromaticity and electronic properties of curved π-conjugated systems has been increasingly attractive for the development of novel functional materials for organic electronics. Herein, we demonstrate an efficient synthesis … Controlling the aromaticity and electronic properties of curved π-conjugated systems has been increasingly attractive for the development of novel functional materials for organic electronics. Herein, we demonstrate an efficient synthesis of two novel wave-shaped polycyclic hydrocarbons (PHs) 1 and 2 with 64 π-electrons. Among them, the wave-shaped π-conjugated carbon skeleton of 2 is unambiguously revealed by single-crystal X-ray crystallography analysis. The wave-shaped geometry is induced by steric congestion in the cove and fjord regions. Remarkably, the aromaticity of these two structural isomers can be tailored by the annulated direction of cyclopenta[b]fluorene units. Isomer 1 (Eoptg = 1.13 eV) behaves as a closed-shell compound with weakly antiaromatic feature, whereas its structural isomer 2 displays a highly stable tetraradical character (y 0 = 0.23; y 1 = 0.22; t 1/2 = 91 days) with a narrow optical energy gap of 0.96 eV. Moreover, the curved PH 2 exhibits remarkable ambipolar charge transport in solution-processed organic thin-film transistors. Our research provides a new insight into the design and synthesis of stable functional curved aromatics with multiradical characters.
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Helical Nanographenes Containing an Azulene Unit: Synthesis, Crystal Structures, and Properties

2019-12-22
Ji Ma , Yubin Fu , Evgenia Dmitrieva +7 more
Abstract Three unprecedented helical nanographenes ( 1 , 2 , and 3 ) containing an azulene unit are synthesized. The resultant helical structures are unambiguously confirmed by X‐ray crystallographic analysis. … Abstract Three unprecedented helical nanographenes ( 1 , 2 , and 3 ) containing an azulene unit are synthesized. The resultant helical structures are unambiguously confirmed by X‐ray crystallographic analysis. The embedded azulene unit in 2 possesses a record‐high twisting degree (16.1°) as a result of the contiguous steric repulsion at the helical inner rim. Structural analysis in combination with theoretical calculations reveals that these helical nanographenes manifest a global aromatic structure, while the inner azulene unit exhibits weak antiaromatic character. Furthermore, UV/Vis‐spectral measurements reveal that superhelicenes 2 and 3 possess narrow energy gaps ( 2 : 1.88 eV; 3 : 2.03 eV), as corroborated by cyclic voltammetry and supported by density functional theory (DFT) calculations. The stable oxidized and reduced states of 2 and 3 are characterized by in‐situ EPR/Vis–NIR spectroelectrochemistry. Our study provides a novel synthetic strategy for helical nanographenes containing azulene units as well as their associated structures and physical properties.
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Electronic transport in polycrystalline graphene

2010-08-22
Oleg V. Yazyev , Steven G. Louie
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The Kinetics of Cellular Proliferation

1960-10-01